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NEW Y ORK LO NDON TORONTO SYDNEY Tomy parents, SaIIyand Leonard, for theirendless loae, guidance,snd encouragement. A universal beauty showed its face; T'heinvisible deep-fraught significances, F1eresheltered behind form's insensible screen, t]ncovered to him their deathless harmony And the key to the wonder-book of common things. In their uniting law stood up revealed T'hemultiple measures of the uplifting force, T'helines of tl'reWorld-Geometer's technique, J'he enchantments that uphold the cosmic rveb And the magic underlving simple shapes. -Sri AurobitrdoGhose (7872-1950, Irrdinnspiritunl {uide, pttet )

Number is the within of all things. -Attributctl to Ptlfltngorus(c. 580-500s.c., Gre ek Tililosopther nrr d rtuttI rc nut t ic inn ) f'he earth is rude, silent, incomprehensibie at first, nature is incomprehensible at first, Be not discouraged, keep on, there are dir.ine things well errvelop'd, I swear to vou there are dirrine beings more beautiful than words can tell. -Wnlt Whitmm fl819-1892,Americnn noet)

Flducation is the irrstruction of the intellect in the laws of Nature, under which name I include not merely things and their forces,but men and their ways; and the fashioning of the affections and of the r,t'ill into an earnest and loving desire to move in harmony with those laws. -Thomas Henry Huxley (1 B 25-1 Bg 5, Ett glislth iolo gis t) Contents

ACKNOWLEDGMENTS xi

GEOMETRY AND THE QUEST FOR REALITY BY JOHN MICHELL xiii

INTRODUCTION xvii

MONAD WHOLLY ONE

2 DYAD IT TAKES TWO TO TANGO 21

3 TRIAD THREE-PART HARMONY 38

4 TETRAD MOTHER SUBSTANCE 60

5 PENTAD REGENERATION 96

6 HEXAD STRUCTURE-FUNCTION-ORDER 178

7 HEPTAD ENCHANTING VIRGIN 221

8 OCTAD PERIODIC RENEWAL 267

9 ENNE AD THE HORIZON 301

10 DECAD BEYOND NUMBER 323

EPILOGUE NOW THAT YOU'VE CONSTRUCTED THE UNIVERSE ... 347

ix fienmetryand thefluestfurfleality

lohn Michell

Sooner or later there comes a time in Me when you start thinking about Reality and where to find it. Some people tel l you there is no such thing, that the world has nothing per- manent in it, and, as far as vou are concemed, consists merely of your fleeting experiences.Its framework, they say, is the random product of a nafural process,meaningless and undirected. Others believe that the world was made by a divine Cre- ator,who continuesto guide its development.This soundsa more interesting idea, but, as skeptics point out, every reli- gion and church that upholds it does so by faith alone. If you are naturally faithful and can accept without question the orthodoxy of your particular religion or system of beliefs, you will feel no need to inqufue further and this book will appear superfluous. It was written for those of us who lack or have lost the gift of simple faith, who need evidence for our beliefs. We cannot help being athacted by the religious view, that the world is a harmonious, divinely ordered cre- ation in whidr, as Plato prornised the uninitiated, "things are taken care of far better than you could possibly believe." Yet superficially it is a place of confusion and chaos, where suffering is constant and the ungodly flourish. ANDTHE OUEST FOR REALIry xlv OEO}IETRY

This is where we begin the quest for Reality' Lgokinq closelv at nature, the firsi insight we obtain is thai, behind the apparently endlessproli{eration of natural objects,there is a flr lesser number of apparently fixed tlpes' We see' for example,that through every generationcats are cats and are the same way, every 'ro"""hu"progrlmmed for catlike behavior. In the unique characteristics of a rosb and every oak these leaf is definitely an oak leaf. No two specimens of -are ever exactlythe same,but eachone is clearly a product of its formative i1pe. If it were not so, if animals and plants,sim- ply inherited their progenitors' characteristics,the order of nature would soon dissolve into an infinite variety of crea- tures, undifferentiated by speciesand kinships' This observation, of one type with innumerable prod- ucts, gives rise to the old philosophical problem o{ the One and the Manv. The problem is that, whereas the Many are visible and tangible ind can be examined at leisure, the One is never seen or sensed, and its very existence is only inferred tfuough the evident effect it has upon its products, the Many. Yet, paradoxically, the One is more truly real than the Many. In the visibte world of nature all is flux' Every- thing is either being bom or dying or moving between the two processes.Nothing ever achievesthe goal of perfection o. ti'," ttute of equilibrium that would allow it to be describedin essencb.The phenomenaof nature, said Plato, are always "becorning," never actually " are." Our fiue sensestell us that they are real, but the iniellect judges dif- ferently, reasoning that the One, which is constant, creative, and ever the sarne, is more entitled to be called real than its ever-fluctuating products. The search for Reality leads us inevitably toward the type, the enigmatic One that lies behind the obvious world of*re Many. Imrnediately we encounter difficulties. Being imperceptible and existing only as abstractions, tyPes can- not be apprehendedby the methods of physical science.A number of modem scientists, perceiving the influence of types in nature, have attempted to bring them within the range of empirical study. Rupert Sheldrake, author of A Nezu Scienceof Life and other works, has taken a bold step in that direction.In an earlier age,the Pythagoreansworked on sys- tematizing the types by means of numerical formulae. Yet SEOiETRY AND TH€ OUESTFOR REALITY

Plato, who wrote at length on the subject of constant types (referred to as "forms"), was carefully ambiguous in defin- ing them and never made clear the means by which they influence the world of appearances. Plato did, however, give instructions on the procedue toward understanding the nature and function of the types. In the Republiche described the ascent of the mind through four different stages.It begins in Ignorance, when it does not even know that there is anything worth knowing. The next stage is Opinion, the stage in which TV chat-show partici- Dants are forever stuck. This is divided into two subcate- gories, Right Opinion and Wrong Opinion. Above that is the Ievel of Reason.By education and study, particularly in cer- tain mind-sharpening subjects,the candidate is prepared for entry into the fourth stage, which is called Intelligence (nous). One can be prepared for it but with no guaranteeof success,for it is a level that one can only achieve on one's own, the level of heightened or true understanding, which is the mental level of an initiate. The studies that Plato specified as most effective in preparing the rnind for understanding are the so-calied mathematical subjects, consisting of number itself, music, geometry, and astronomy. These were the main studies of Plthagoras and his followers, who anticipated the realiza- tion of modem physics in proclaiming that all scales and departrnents of nature were linked by the same code of number. Geometry is the purest visible expression of num- ber. In Platonic terms, the effect of its study is to lead the mind upward from Opinion onto the level of Reason,where its premises are rooted. It then provides the bridge or ladder by which the mind can achieve its highest level in the realm of pure Intelligence. Geometry is aiso the bridge between the One and the Many. When you draw one of its basic figures-a circle, say, or a triangle or regular polygon-you do not copy someone else's drawing; your model is the abstract ideal of a circle or triangle. It is the perfect form, the unchanging, unmanifest One. Below it are the Many-the expressions of that figure in design, art, and architecture. ln nature also the One circle gives rise to the Many, in the shapes and orbits of the plan- ets, in the roundness of berries, nests, eyeballs, and the xvi GEO}IETRYAND THE OUEST FOR REALITY

cycles, of time. On every scale, every nafural pattern of growth or movement conforms inevitably to one or more of the simple geomerrictypes. The pentagoryfor example,Iies behind each specimen of the five-pelaled rose, the five_ fingered starfish, and many other living forms, whereas the sixfold, hexagonal gpe, as seen in the structure of snowflakes and crystal generally, pertains typically to inan_ imate nature. As soonas you enter upon the world of sacred,symbolic, or phiJosophicaigeometry-from your first, thoughifuJcon_ struchon of a circle with the circumferencedivid;d into its natural six parts-your mind is opened to new influences that stirnulate and refine it. you begin to see,as never before, the wonderfuily patlerned beauty-of Creation. you see true artistry, {ar above any human contrivance. T?risindeed is the very, source of art. By contact with it your aestheticsenses arc heightenedand set_uponthe firm bisis of truth. Beyond the oovlous pteasureot contemplatingthe works of nafure_the MTy-1" the delight that comes through the philosophical study of geometry, of moving toward-the prr".,." of th" One. Michael Schneideris an experiencedteacher and, as you areentitled to expect,a masterof his geometriccraft. No Lne less qualified could set out its basic piinciples so ctearty anJ simply. His much rarer assetis appieciation of the symbolic and,cosmologicals].'rnbolism injrerent in geometryiThat is the best reasonf9r being interestedin the subject,and it is why. rhe.philosophers 1"_,t:i:." of ancient Gieece,Egypt, and other civilizations made geometry and numbei'the most important of their studies. The traditional science taught in their mystery schoolsis hardly known todav.Itis not available for study in any modem place of educition, and the.reis very little writing on the sublect.ln this book you will find something that cannot be obiained elsewhere, a complete introduction to the geometric code of nature, written and illustrated by the most perceptive of its modem investigators. lntrndurtim

FANNINGANCIENT SPARKS The universe may be a mystery, but it's not a secret. Each of Thr*ort us is capable of comprehending much more than we might incomprehensible thing realize. A vision of rnathematics different from that which aboutthe uniaerse is that we were taught at schoolholds an accessiblekey to a nearby world wonder and beauty. it is comprehensible. of In ancient Greecethe advanced students of the philoso- -Albert Einstein (1,879-1955) pher Pythagoras who were engaged in deep studies of natural science and self-understanding were called, mathe- matekoi,a'those who studied all.l' The word mathemasignt- fied "leaming in general" and was the root of the Old Eng- lish mathein, "to be aware," and the Old German munthen, "to awaken." Today,the word math has, for most people, constricted its scope to emphasize mundane measurement and mere manipulation of quantities. We've r;nwittingly traded wide-ranging vision for narrow expertise. It's a shame that children are exposed in to numbers merely Nothing educationis as quantities instead of qualities and characterswith distinct soastonishing as the personalities relating to each other in various pattems. If amountof ignoranceit only they coirld seenumbers and shapesas the ancierrtsdid, as s)zmbols principles accumulatesin the form of of availableto teachus about the nat- inert urai structure and processesof the universe and to give us facts. perspective on human nature. Instead, "math education" for -Henry Brooks Adams children demands rote memorization of procedures to get (183&-1918,American one "right answer" and pass innumerable "skill tests" to historian) prove superficial mastery before moving on to the next iso- lated topic. Teachers call this the "drill and kill" method. Even its terrninology informs us that this approach to math is full of problems. It's no wonder countless people are irmu- merate. We've lost sight of the spiritual qualities of number and shape by emphasizing brute quantity. XYIII

This book concemsmathematics, but not the kind you were shown at school. The Roman goddessNumeria is said to have assistedin teaching each child to count' We must have had a misun- derstanling becauseI grew uP on uneasy terms with the subject of irathematicJin schbol. I was intrigued b.y.the Bodily ,rrrcise,when inJinity of numbers and could calculate mechanically if I had taught for a given situation' compulsory,does no harm memorized the rule I was Although I liked science and art, math intimidated me' I but knowledge to the body; remem6er my frustrated tears at age seven over the concePt tohichis acquiredunder of subtracting by borrowing {rom another column when it compulsionobtains no contained a zero.l dreaded math, its mindless memoriza- who have holdon themind. tion and its tedious paperwork' (Pify the teachers to check it!) Math was dry and mechanical and had little rel- -Plato (c. 428-348e.c., Greek evanceto my wo d. If we had iooked at numbersto seehow mathematical philosopher) they behave with each other in wonderful patiems I might have liked math. Had I been shown how numbers and shapes relate to the world of nature I would have been ihdlled. hstead, I was dulled by math anxiety and pop ottizzes. Fortunately for me, when I was sixteen one of my teach- ers mentioned that mathematics can be found in nature: a Wrao* shouldbe six-sided snowflake is shaped like a bee's cell and quartz crystal. I was stunned to make the cormection. How could cherishedas a meansof something as apparently irrelevant as mathematics be traaelingfrom youth to old related to something as wonderful as nature? The ordinary age,for it is morelasting world opened up to rne and spoke the language of number thanany otherpossession. and shape. No longer a foe, the dreaded mathematics becameat once a teacherand a tool. Nature wasn't what it -Bias of Priene (c.570B-c-, was made out to be, an antagonist to fear, conquer, and one of the SevenSages of exploit, but a garden of wonder and a patient teacher wor- ancient Greece) thy of gteat respect. Over the years rny studies led me to seethat a profound understanding of number was prevalent in ancient times, more than is commonly acknowledged, and seamlessly wove mathematics, philosophy, ar! religiory myth, nature, science, technol oW, and everyday life. This book is a fan- ning of some little sparks of philosophy from deep antiquity to introduce ihe general reader to another view of mathe- matics, nafure, and ourselves that is our heritage and birthright. It requires that we liberate math from its Pigeon- xix

Cqlvinona Hobbes by Bill Wotterson

'(EAl{. A\! 1tl6E EAUAnoN: IHls ul\{nE Bcd( ts N\! AF€ L\KEMIRAC,LES, \C'\) OFII{INGS THAT HA\IE lb IAG TTION\}USgS A\O iI\ITN E. A<€PIF' OI\ FAII\{I YCNAOD l{EM, T€I MAGI(A$ IT5 A. &cch4EoNE r0ty NUMER! REL\GIONI NOONE CAN SA\ I{ON If HARENS.\C|\} EIII.$R EEL\BIE r oR.\cN Dor\T. I M, /vt-, hole and see it spanning the framework of the most inter- disciplinary topic possible:the uliverse. Plato wrote that all knowledge is already deep within us, so no one can really teach us anything new. But we can remind each other of the archetypal principles of number and nature we already know but may have forgotten. This book is intended as that reminder to guide you through this world of wisdom, beauty, uplift, and delight and to remind By *rr* [theGreeks] you of the gentle, wise principles by which the universe is designed. geometrywas held in the aeryhighest honor, and nonewere more illustrious THREE LEVELS OF MATHEMATICS than mathematicians. B ut Mathematics is whole, but it rnay be divided into three lev- we [Romans]haoe limited els or approaches:secular, symbolic, and sacred. thepractice of this art to its usefulnessin. Secutor Mothemotics measurementand What is taught in schoolcan be called secularmathematics. calculation. Adding the amounts of a storc purchase,,calculating change, weighing produce, measuring ingredients for a recipe, -{icero (106-43 0.c., Roman counting votes, telling time, designing a bookshelf or sky- orator, statesman, and scraper, measuring iand boundaries, stock market econom- philosopher) ics, calculating tax: we're taught ihese are the ends of math- ematics for nonscientists. We train children to be hurnan pocket calculators.Even many educated people who are adept at calculating have no idea that there is more to math- ematics than mere reckoning of quantities. This quantitative approach keeps us duII to the potential wisdom that the familiar counting numbers can teach us. \ltrhen imaginativeiy taught to people begiming at an early age, mathematics can delight, inspire, and refine us. It can make us aware of the pattems with which the world and we Ilature, thatuniaersal are made. Instead, math is taught as a servant of co [nerce, andpublic manuscript. without regard for its basis in nature. It is viewed as a dis- tant subiect that instills much more anxiety than wonder -Sir Thomas Browne and inspiration. Mathematics is seen as outside us to be (1605-1682,English physician occasionally caiied upory rather than woven into the fiber of and author) our existence.

SUmbotic Mothemotics Nature is writtenin Embedded in the very structure of ordinary numbers is symbolsand signs. another level, which may be called "symbolic" or "philo- sophical mathematics." It is well known that simple num- -John GreerLleaf\A4rittier bers and shapesrelate to each other in harmonious recurring (1807-1892,American poet) patterns. Just notice how floor tiles containing different shapes mesh. Mathematicians and scientists seek and study patterns as clues to a deeper understanding of the universe. What's more, s).'rnbolic mathematics recognizes mrmbers and shape patterns as representative of far-reaching princi- ples. They can be guides to a deep cosmiccanon of design. Nature itself rests on an intemal foundation of archetypal principles symbolized by numbers, shapes, and their arith- metic and geometric reiationships. AII thingsare full of According to ancient mathematical philosophers, the signs,and it is a wise man simple counting numbers from one to ten and the shapes whocan learn about one that represent them, such as circle, line, triangle., and square thingfrom another. express a consistent, comprehensible language. The ten numbers are a complete archetypal sourcebook. They are the -Plotinus (205-270, Roman original ten patents for designs found all tfuough the r.rni- Neoplatonic philosopher) verse. These ideal oattems are the ones that were skewed and veiled in schobl and that nature approximates in all transitory forms, from the smallest subatomic particles to largest galactic clusters, crystals, plants, fruits and vegeta- bles, weather pattems, and animal and human bodies. Any- INTRODUCTION xxl

Islamic tiling patternof a mosque and structure of the boric acid molecule show how identical shapes inter­ lock in defined, recurring patterns in both art and nature.

J am not ambitious to appear a man ofletters: I could be content the world should think I had scarce looked upon any other book than that ·ofnature.

-Robert Boyle (1627-1691, British physicist and chemist) thing anyone can point to in nature is composed of small patterns and is part of larger ones. Jn nature's infinite book of Historically, nature has been compared to a book secrecy, a little I can read. written in geometric characters. .... Reading the Book of Nature first requires familiarity -William Shakespeare (1564-1616) with its alphabet of geometric glyphs. We're exposed to nature's text in the natural shapes around us, but we don't recognize it as something we can "read." Identifying shapes and patterns and knowing what principles they represent allows us to understand what nature is doing in any given situation and why these principles are applied in human affairs. Why are plates and pipes and planets round? Why do hurricanes and hair-curls, atoms and pinecones unfurl as spirals? Why xxii

didn't school teach us that when we see the same spiral shape in shells, galaxies, and watery whiripools we are wit- nessing the principle of balance through motion? Ol that the hexagonal cells of beehives package the maximum space using the least materials, energy, and time? Nature labels everything with a cosmic calligraphy, but we generally don't suspect even the existence of the language. It is an open For thenature of number secret, fully in view but usually ururoticed. Like consonants is to beinformatioe, and vowels-like building blocks and Srowth Patterns- numbers, shapes,and their patterns s;'rnbolize omnipresent guiding and instructiaefor principles, including wholeness, poladry shucture, balance, in eaerythingthat anybody cycles, rhythm, and harmony. Each shape represents a dif- is subjectto doubtand thnt ferent problem-solving strategy in the cosmic economy. To is unknown. see more deeply into this design alphabet we must be con- For nothingabout versant in nature's native tongue, the language of symbolic mathematics. This book is primarily concerned with sym- thingswouldbe bolic mathematics. comprehensibleto anybody,neither of things Socred Mothemotics in themselaes,nor of one in The terms sites,""sacred geography,""sacred archi- relatian to the other,if "sacred tecture," "sacred arithmetic," and "sacred geometry" seem numberand its essence overused today. To the ancients the "sacred" had a particu- werenonexistent... lar sigrrificance involving consciousnessand the profound Theessence of number, mystery of awareness. How are you and I aware of these Iike harmony,does not very words and their significance? Now that you've read them, these words are no longer just on the page. They're allow misunderst anding, within your awareness.Sacred space is within us. Not in out for this is strangeto it. body or brain cellsbut in the volume of our consciousness- Deceptionand enr'y are \Arhereverwe go we bring the sacred within us to the sacred inherentto theunbounded, around us. We consecratelocations and studiesby the Pres- ence of this awareness,not just the other way around. \44ry unknowable,and should the sites of stone temples and wonderful cathedrals unreasonable.. . kuth, be more sacredthan a rocky desert or concretecity streetif howeuer,is inherentin the we bring holy consciousnessto each of them? Geographic natureof numberand locations are no more or less sacred than any other, although is inbredin it. they may be powerful telluric sites. Awareness the ulti- mate sacred wonder. \4rhy endow objects outside ourselves -Philoiaus (Fifthcentury as sacred and ignore the same source within us? A surpris- e.c.,Greek Pythagorean ing amount of the world's religious art and architecture has philosopher) been designed using the timeless symbolic pattems of xxiii

nafure and number, but these pattems remain syz bolicof our own sacredinner realm, s).'rnbolic of the subtle structure of awareness whose source is the same as archetypal number- All this was understood in ancient times ani'deemed so important that ii was built into the culture on every level. "You Modem science tells us that what we comrnonly call amuseme," I said, "rcality" is a compilation of pictures based on u ,rirro* "with your oboiousfear sense-bandview of surface features. The world we perceive thnt thepublic will is a small slice of a vast, mostly invisible energj,-eve.,t. Mathematics can take us beyond our ordinary linriis to the disapprooeif the subjects cosmicdepths. Plato at his Academy requirej the studv of you prescribedon't seem mathematicsas a prcrequisitefor pftito{oinA, u term sigriify- usefal.But it is in fact no ing "the love of wisdom" and lift "to the soul to truthl Iu;t easymatteL but ztery over a centuq/ earlier Pythagoras had invented the word dfficultfor "p_hngsophy" as a result of a question posed to him. I4rhen peopleto asked "Are you wise?" he is said to have answered ,,No, but belieaethat thereis a l'n:. pythagoras a loaer of wisdom." Both and plato sug_ faculty in the mind of each gested that all citizens leam the properties of the first ten of us which thesestudies numbers as a form of moral instruction. The study and con_ purify and rekindleaffer templation of number and geornetry cnn show us, if we look it wi$ the eyes of ancient mathematical philosophers, that hnsbeen ruined and neither outer nature nor human nature is the hodgepodge it blindedby otherpursuits, may seem. Symbolic mathematics provides a map of our though it is moreworth own irurer psychological and sacred spiritual strucir.re. But pleserving than any eye studying number properties and intellectually knowing the road map, the symbolism, is not the sameas achrallv ta*kine sinceit is theonly organ the joumey. We take that joumey by finding within ourl by which we perceioethe selves the universal_principles these properties represent truth. Thosewho agree and by applying the knowledge to our own growth. We pay with us aboutthis wiII gioe to paying attentiory in imageless u*u.".i"rr, lltention y our p directing sustained attention to that whici the srrmbolsrefq ropo s als unqualifie d to within us. lA/henthe lessonsof symbolic or philosophical approaal,but thosewho mathematics seen in nature, which were designed into reli_ are quite unawnreof it will gious architecture or art, are applied (not functioiatty iust probablythink you are inteffectually) fo facilitate the growth ani transformitiin of ion- talking nonsense, sciousness,then mathematicsmay rightly be called ,,saired.,, as they ,,sacred To me, the terms "sacred arithmetic;and geometry,, Toon'tsee what other onJyhave significancewhen grounded in the exieriencetf benefitis to beexpected selJ-awareness.-Religious art is sacrednot only due to its from suchstudies." subject matter but also becauseit was desigrredusine the subtle s).'rnboliclanguage of number,shape, Ld propoition xxiv INTRODUCTION

to teach seLf-understanding and functional self-develop- Th, ,rd mysteryof life is rnent. Ancient Egyptian arts, crafts, and architecfure Per- not a problemto besolaed, haps provide tnJ"Uest accessible examples of design that geometuy, and nature to it is a realitYto be .rc"d itt symbolism of nurnber, teachan acceleratedform of self-developmentto trained ini- experienced. tiates who knew how to translate the symbolism into medi- tative exercises. -J. J. Van der Leeuw Thus, true sacred Seometry cannot be taught through books,but must .u*uii asputi of the ancientoral tradition passed from teacher to pupil, mouth to ear. Becauseover ienturies this knowledgewas passedalong secretlyso asnot to conflict with the prevailing intolerant religious authori- ties or be disposed to those considered profane, there is still Th, gootof tifeis liaing in an aura of mysticism about it. But nature's Patterns and agreementwith nature. those o{ our inner li.fe are familiar to everyone and always available to us. The power which we seek is the p ower with -Zeno of EIea(490-430 8.c., which we seek. When we feel separate from the archet'?es Greek philosoPher) of nature, number, and shape we make them mystical, but this orilv keeps our selves in a mist. There is no need for ';occultism" secrecyand any longer. These are everyone's iife-faits. We can apply them to better appreciatethe world, and we'Il need thern once we realize the urgency of cooper- ating with the way the world works. This book is concemec W, orrbornfor with dispelling the mysticism surrounding sacred mathe- cooperation. matics by reiniegrating the timelessknowledge of number and shape wi*i its familiar expreisions in nature, art, and its -Marcus Aurelius(121-180, -practical significancefor self-development. philosopherat twelve, Studying, contemplating, and iiving in agreement with R6manemperor at forty) universal principles is a social responsibility and can be a spiritual path. It is becoming clear that when we cooperate *ith nature'sways we succeed;when we resist,we struggle. Imolications for our environmental crises are obvious. Rather than an antagonist, nature can be our teacher to leam from and cooperate with to mutual benefit. To understand nature better,we first need to recognizethe roles of its basic pattems.

MYTHMATICS:MATH AS CREATIONMYTH To say that ancient seers viewed the universe and them- selves in a way different from ours is understatement. Today, we are emerging from the grip of literalism, mere INTRODUCTION xxv

quantitative measurement, and analysis. The ancients had a more poetic and synthesizing vision. We think we invented mattu but the ancients knew that they were involved in a process of discovery. They personified in stories the forces of nature and gavethem namesof gods,goddesses, and nature spirits. Mythological exploits heiped explain the orisins and ways of familiar events from world creation t"o night an{ day, storms, agriculture, wars, love, hate, joy, sleep, iife and. death, and the full spectrum of naturai una ni -ur, aftars. ttut today we condescendand say that,show, in their childlike way, those pagans attempted to explain the mvs_ ,,enlightened,, teries of the world while lacking our scientific understanding. We think we rel y underitand the forces of nature becausewe have labels,theodes, terminology, exper_ iments, measurements, advanced formulas, and aJtoundino Divinity cfucumscribing technological the tools. With our habit of sensitle h,".ulir- *E limits of the universe.ilote have a difficult tirne believing that the ancient mfihopoeic his foot beyond the frame. imagination could have been a valid tool for seriou" kno*f (BibleMorulisee, c. 1250) edge. Yet it is we who have the cruder understanaing. Uy pigeonholing fraglnented facts we miss the whole haimo_ ruous picfure. We grope in a world we consider dangerous, accidental, and chaotic but one that is actually harm"onious and awaiting our cooperation. If only we could seewith ihe eyesof the ancients! Scientists confiim with formulas what ancient seers knlw thrgugh revelation: that the worta,s puttems- and cycles are harmonious when seen as mathemaiical relation_ ships. The ancientsi intuitive revelations were grou.a"aiy "ff," the study of nature, numbers, and shapes. u".i"", werS particularly !,1eks interestedin thi ways g"o;uti. shapesmesh. in. recurring harmonious pattems (ihe ,,fitting Greek w3rd harmoniasignifies togethe/,), aswe seein floor tiles, quilts, and wallpaper patterns, and how forces of nature orchestratein small and large cycles(as satellites now o.l" "patLrn).'Wh; ::y:ul worldwide weather ;;-;" "things," nouns and discrete objec'ts,the ancient rnathe-mli ical phiJosophers sawprocesses, verbs, transformine outi;;" mesheclharmoniously. ,,buck,, ln truth, the never s"tops.The r-ropt .ranguageretains this vision, having no nouns.(i,m ,,I,m not The Ancient of Days design- "wearing a shirt,' but shirting.,,) 'cosmos" iikewise, ;.;;rl mg the universe through generally refers to ,,oute-r space,,,Bui the word geometry. (Ro&efeller Center) derives trom the Greek kosmos(signi$ng,,embrcjdery,,), xxYl INTRODUCTION

which implied not a universe like a huge room fi1led with discormected noun-things but the orderliness and harmonyof woven Datterns with which the universe is embroidered and moves. Kosmossignified the honorable and "right behavior" of the whole, the harmonious orchestration of the world's pattems and processes.In this original senseour word "cos- metics" refers not to the nouns involved, the lipstick and rouge, but to the process of bringing the elements of the face Our biggestfailure is into harmony. By studying the recurring harmonious pattems inherent ourfailure to seepatterns. in mathematics, music, and nature, ancient mathematical -Marilyn Ferguson philosophers recognized that consistent correspondences (American writer) occur throughout the universe. The Greeks investigated ariihmetic by arranging pebbles in various geometric shapes ("figurate arithmetic") and so rmcovered the arche- typal pattems and principles inherent in many tyPes of le1a- tionships. This sfudy developed over centuries from what we call "number magic" to become the modem branch of mathematics called "number theory." Through extensive

TRIANGULAR NUMBERS a a ao ooa aaa a ao oaa aaaa a aa aao aoaa aaaaa 13610 15 +2 +3 +5 SQUARE NUMBERS aaaoa ooaa aaaoo Figurate Arithmetic. Simple aoo aaaa aaaao a angements of pebbles oa aaa oaaa aaaao showed ancient mathema- O Oo OOO aaaa aaaoa ticians the pattems inherent in relationships among num- 149 l6 25 bers. For example, any two +3 +7 +9 consecutive "triangular" nurnbers(1, 3, 6, 10,15 . . .) a oa added together always make a aa a "square" number (1, 4, 9, aa aoa 16,25...). 3+ o- xxvii

study of patterns and by intuitive revelation the ancients realized that the structure and pattems of arithmetic and geometry reenact the creating piocesses found ali through nature. In both number and nature they saw the same divine impress. Strip away all the sensory characteristics of color, texture, tone, taste, and smell from an obiect and onlv num- ber remains as its size,weight, and quantity. Takeaway the leatures associatedwith number and it's all gone. Ordinary numbers and shapes represent eternal verities in a form made comprehensible to us. Mathematics is a philosophical language that reveals the Greek archai, tlrrefowtdatlon prin- Theharmonyo|theworld ciples of universal design and construction. Numerals-, the is mademanifest in Form written s).'rnbols of nontangible numbers, and geometric andNumber, and theheart shapes,are the emblems of these archetypal principles. The and soul archetypes are universal in that they are the iame io every- andall thepoetry one everywhereand in every era ofN atural Philosophy are Nature's . harmonious principles are exhibited through embodiedin theconcept of its mathematical relationships. As we leam to read its archl_ mathemnticalbeauty. typal language, we discover that the topic of the book of nature is a story a my.thicquest about thl processof trans- -Sir D'Arcy Wentworth formafion. This mathematjcalmyth of the ireative process. Thompson (1860-1948, this "mythmatics," is not ancie* or new-age but timeless, Scottish zoologis! classical accessiblein everyrage becauseit encodesetemal constants scholar) available to _all.Although the m1th,s outer form adapts to different cultures, it may be rediscovered in any eia bv exarnining simple numbers, basicshapes, and evei-present nature. Thus, people of every historical period can under_ stand the principles of nature,s creating processby imagi_ natively_examining the corresponding arihetypal relatioin_ The world,harmoniousl y ships inherent in mathematics, personified in mythology, confused, and depicted in art and culture. This book is intended a;; Whereorder in begirmer's guide to the process of recovering that vision. oariety we Lr the view of philosophical mathematiiians, numbers qP2 and their associatedshapes represent stages in the process And where,tho' all things ol becoming.Ifhile integrated in the whole, eachhis a life dtfft , ot rts own and a unique role in the cosmicmyth. The arche_ AII agree. types of number and shapewere personified in ancientcul_ tures as various gods, goddesses,and world_builders. The -Alexander Pope classicmyths eJaboratedon particular aspects of the univer_ (1588-1744,English poet) sal principles they representbd. Their intirrelationships and ,,soap liaisons, which we consider a opera,,, transmitted xxvlii

timelesstruths. Though statesanction, as in Egypt, Greece, India, Africa, Chin4 Tibet, Native America, and elsewhere, all aspects of society were organized according to a canon consistent with nature's own structure. The proportions of temples were always designed, situated, and built in accord with the number and shape symbolism representing the An *rr rXrrr, ofNature ternple's deity. Through the process known as " gemaftia," are only the mathematical which associatesnumber values with letters of a carefully constructed alphabe! as in Greek, Hebrew, Arabic, Syriac, consequencesof a small and Egyptian, the names, titles, and atkibutes of deities numberof immutablelaws. reveal, to those initiated to the code, their role in the cosmic constructive process. For example, seven was known as the -Pierre Simon de Laplace number of the "virgin" becaule no number below seven (1749-1827, Frcnch enters into (divides) it, nor does it "reproduce" another astronomer and number within the first ten. Through gematria, the mathematician) Gieek letters of the name of the maiden goddess Athena add up to seventy-severy giving us a standard for measuring length, distance. The letters of her epithet, Pallas, add to 343 (which equals 7 x 7 x 7), indicating volume. The letters of her appellanonparthenos("vftgSn") add to 515, and it is no coin- Toknow all,it is cidence that 51.5 degrees is extremely close to the angle necessaryto knowaery within a regular heptagon, the shape with seven equal sides Iittle;but toknou.t that and angles, giving us area. Each "deity" represented a set of oerylittle, one must archetypal principles that were rnade cornprehensible as first mathematicalpattems for practical use as lengih, area,vol- lcnowpretty much. ume, weight, duration, and-the music and were applied to the --€eorges architecture and ritual of deity's temple precinct. Each I. Gurdjieff deity (1.872-1949) was responsible for overseeing part of the world,s har- monious structure in the same way that his or her mathe- matical pattems did. Ancient mythology was consciously integrated wi*r, and symbolized by, the universal and time- less canon of nafure's mathematics. This book makes practical the teachings of the Pythagoreans and others (including Plato, Theonbf Smpna, Iamblichus, Philolaus, N umbersare the highest Proclus, and Nichomachus) concem- ing the numbers one to ten as a complete frame for explor- degreeof knoutledge.lt is ing and constructing the fundamental forms of the univirse; knowledgeitself. These fust ten numbers are like seedsfrom which all subse- qrrert numbers and shapes grow sharing and expanding -Plato their properties. The use of Greek words for these numbei xxtx

princrples, Monad, Dyad, Triad, and so on, as chapter titles, is intended for rnore than numerical order. These terms rep_ Man, thoughseeing, resent a processof unfolding cosmic qualities and prinii_ suffered blindness.. . ples. Each ftom chapter is devoted io one number and shapi prin- And,forhim,Ifound ciple. Although we count in sequencefrom one to ten, irese Numbers,the number principles do not only unfold sequentially but inter_ purestof penetrate the universe simultaneously-in a coimic svm_ inoentinns. phony. -Aeschylus The cosmic creating process is deep within us and can (52F-456n.c., Greek tragic emerge through our hands with the heip of the three tradr- dramatist) tional tools of the geometer-the compais, the straightedge, and the pencil. To the ancients these tbols represent""dthr"". divine attributeS that designed the pattemi of the world. ust ln ttds century J modern scienceis rediscovering-their what the G eometryis lotozoledgeof ancients s)rmbolized by these three tools and use. theeternally existmt. Albert Einstein referred to them as E = mc?,the famous rela_ ,,tools,, tionship among the universe,s three of the -Plato rating process:light, energy,and mass(or matter).The "orfier_ ro'ies and motions of the geometer,sthree tools in geometriccon_ squction replicate the universe,s o-., p.o""ss whereby i

timelessnessof nature's creating process by consciously -participating in geometricactivities. By-doing geometric constructionson paPet or with a stick in the sind, we are recalling the agelessprocess of cre- ation, replicating with our mind and hands the generative principles by which the universe is evolving. Our construc- iions wiil dLmonstrate to us the harmonious princiPles at work in the world. T mathematics encode subtle experi- If you do not restupon the S)'rnbolic and sacred encei whose purpose is different from that of secularmath- of nature goodfoundation , ematics.They caninvigorate, refine, and elevateus. Our role you will laborwith little as geometeriis to discoverthe inherentproPortiory balance, honorand lessprofit. utd hut-ot y that exist in any situation. The study and experienceof numeric and geometricproportion infuses in [T]hosewho take for their uJan appreciationof proportion everywhere.The study of standardany onebut balanceteaches us to recognizeand seeka senseof baiance nature-the mistressof all in our lives. The study of harmony develops our sense ot ffiastefs-IDeary harmony in all relationships.Actually to seeand work with themselaesin ztain. unity and wholeness in geometry and natural forms, rather than iust read about them, can help abolish our false notion -Leonardo da Vinci of separatenessfrom nature and from each other. It is this (1.4s2-1519) notion that ultimateiy fuels competition for the "goods o{ the earth" and contributes to environmental crises. The material of this book developedover years of study and through holding public workshopsand coursesfor peo- ple of widely varying interests,especially artists, architects, and educators. \Alhen rny interest in this subject first began, I was sure that someone must have written one book explaining to my satisfaction the relationship between num- bers, shapes,nature, science,art, and se1f.The information seemsso fundamentai, and people have been around for so many millennia to view it, that I was surprised to learn it only existsin books, articles,and fragmentswritten mostly for specializedinterests. I couldn't find the one book I was looking for that tied it a1ltogether, so I wrote it. It seemsthat the best approachto show you what I've leamed is to take you beyond reading and involve you in the already-existing sy'nthesis of geometry, natural science, art, and sel{-under- standing through a "hands-on" approach. This book is intended to facilitate these studies in such a way as to see xxxl ourselvesas part of the overall harmony and have the tools to engagein it. It will guide you:

r To seehow number and shape srrmbolisrn is already familiar to us through popular sayings, fairy tales, myths, and religious ritual. e To examinethe archetypalprinciples expressed NAruRE HASAU^'A/S by stmptenumbers and shapesand to let vou BEEN A rf]'SlERf/ verify the natural relationshlps existing arnong them. You'll do this by following step-6y-step" instructions for re-creating the basic geometric constru.tions s)'rnbolizing unfolding universal principles. You will use the tools of the geometer,s creating process-the compass,straightidge, and penc -to make two-dimensional and three_ dimensional models of the principles expressedby the numbers from one to ten. . To seewhere, and understand whv, these numericaI archetypesprecipitate as organic and inorganic forms in nafure and are the ones we relv upon in tectmology and engineering. you will be shown how to apply the geometric constructions to images of familiar natural forms discoverins for yourself the ways nature speaksh", g.o-"ii" The real aoyageof language. discoaeryconsists not in . To examine examples of art throughout the seekingnew lands but world that display mathematicaldesign. By seeingroith new eyes. applying the same geometnc conskuctions to reproductionsof art in the book (using tracing -Marcel " Proust (1.871.-1922, paper), you will leam to seehow arch-etvoal French novelist) mathemaficalprinciples have beenpurposefully and.consistently applied, to a surprisAg aegrei to jewelry, art _ and architecture throughoul thJworld, endowing them with powerful coripositional structure,spnbolic significance,unJ intended psychological effects. Sorneexamples will show the geometry the artist used.Further examplesare left for you to complete and explore based on the geomehic constructions you have learned. )m(|l INTRODUCTION

This is not school, so doodling here at any time is okay, even encouraged. The word "doodle" signifies absent- minded scrawling and is related to "dawdle," wasting time in inconsequential activity. Both words derive from the Low

Vomef--, into thelight ffnil3J31',:o,o,Jff:i'il?'bxi::,':lfiT:.:Tj.Til1l forthofthings, 3ff"*ffi:il.T1ff"J?;fffffi:f;:iHffi"#i:ili Let Nature be your teacher. aware of your doodles in a differenl way. psychologistsial doodles manifdstations of the unconscous. Letting our pen- -Willian Wordsworth cil flow without conscious intellectual control allows the (177G-1850, Englishpoet) archetypal pattems from within us to emerge. It is a geome- ter's method for self-discovery. But don't just take my word for this. Try everything yourself. Keep as your own only what your experience val- idates. As legal residents of the cosmos we have the authority to view the worid's blueprints. Here they are.Construct thi pattems that construct the universe.